Question: $\dfrac{ 5k - l }{ -8 } = \dfrac{ k + 4m }{ 9 }$ Solve for $k$.
Solution: Multiply both sides by the left denominator. $\dfrac{ 5k - l }{ -{8} } = \dfrac{ k + 4m }{ 9 }$ $-{8} \cdot \dfrac{ 5k - l }{ -{8} } = -{8} \cdot \dfrac{ k + 4m }{ 9 }$ $5k - l = -{8} \cdot \dfrac { k + 4m }{ 9 }$ Multiply both sides by the right denominator. $5k - l = -8 \cdot \dfrac{ k + 4m }{ {9} }$ ${9} \cdot \left( 5k - l \right) = {9} \cdot -8 \cdot \dfrac{ k + 4m }{ {9} }$ ${9} \cdot \left( 5k - l \right) = -8 \cdot \left( k + 4m \right)$ Distribute both sides ${9} \cdot \left( 5k - l \right) = -{8} \cdot \left( k + 4m \right)$ ${45}k - {9}l = -{8}k - {32}m$ Combine $k$ terms on the left. ${45k} - 9l = -{8k} - 32m$ ${53k} - 9l = -32m$ Move the $l$ term to the right. $53k - {9l} = -32m$ $53k = -32m + {9l}$ Isolate $k$ by dividing both sides by its coefficient. ${53}k = -32m + 9l$ $k = \dfrac{ -32m + 9l }{ {53} }$